
Egwald: Topics in Astronomy and Cosmology The Curvature of Space
by
Elmer G. Wiens
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The Curvature of Space
Introduction
The cosmological principle states that the Universe is homogeneous and isotropic on large scales. Three cosmological models of the spatial
constituent of the Universe that conform to this principle are hypersurfaces in a 4dimensional space with uniform positive, zero, and negative curvature.
The space of each model is specified by systems of coordinates that locate an object, and by a metric that determines the distance between objects
3D Euclidean Space: Zero Curvature (Κ = 0)
Figure 1: An object can be located by its Cartesian coordinates, (x, y, z), or its spherical coordinates,
(r, θ, φ), 0 <= r < ∞, 0 <= θ <= π, 0 <= φ <= 2 π.
An objects position vector is:
l = (x, y, z) = (r*sin(θ)*cos(φ), r*sin(θ)*sin(φ), r*cos(θ)).
Its vector of differentials is dl = (dx, dy, dz), and the space’s metric is the dot product:
dl*dl = dl² = (dx² + dy² + dz²) = dr² + r² [dθ² + sin²(θ) dφ²].
Setting ψ = r / R, this metric can be written as:
dl² = R² (dψ² + ψ² [dθ² + sin²(θ) dφ²]).
4D Euclidean Space: Hypersurface with Positive Curvature (Κ = +1)
Figure 2: An object’s position vector is:
l = (x, y, z, w) = (α*sin(θ)*cos(φ), α*sin(θ)*sin(φ), α*cos(θ), η*cos(ψ)), where α = η*sin(ψ)
is the projection onto the 3D Euclidean space. These coordinates satisfy:
η² = (x² + y² + z² + w²), and 0 <= ψ <= π, 0 <= θ <= π, 0 <= φ <= 2 π.
The Waxis is perpendicular to the 3D Euclidean surface. The radial vector η with length η at the
polar angle ψ is projected onto the 3D surface as the vector α with length α. The coordinates
of η satisfy η² * cos²(ψ) + η² * sin²(ψ) = η².
The area of the yellow sector is η^{2} * ψ / 2.
Figure 3: The radial vector α = (x, y, z) with length α is projected onto the 2D Euclidean plane
at the polar angle θ as the vector β = (x, y, 0).
The 4D space's metric is: dl² = dη² + η² dψ² + η² sin²(ψ) [dθ² + sin²(θ) dφ²]. On a hypersphere of
radius R with dη = 0, η² = R² = x² + y² + z² + w².
The metric: dl² = R² (dψ² + sin²(ψ) [dθ² + sin²(θ) dφ²]), where 0 <= r <= R * π.
Figure 4: The projection of the hypersphere onto the ZW plane, where θ = 0, x = y = 0.
Since ψ is the angle between the Waxis and the radius vector, the distance along the ψ coordinate geodesic is r = R * ψ.
The (z, w) coordinates satisfy: z² + w² = R² * sin²(ψ) + R² * cos²(ψ) = R².
The projection of the point on the sphere onto the Zaxis is α = R * sin(ψ).
With R = 8, and ψ = 0.6981 these values are:
z = α = R * sinh(ψ) = 5.1421; w = R * cosh(ψ) = 6.1285
r = R * ψ = 5.5848 = the distance along the ψ geodesic = the line integral of the curve traced by the radius vectors from 0 to ψ.
Yellow Area: R^{2} * ψ / 2 = 22.34


4D PseudoEuclidean Space: Hypersurface with Negative Curvature (Κ = 1)
Figure 5: An object’s position vector is: l = (x, y, z, w) = (α*sin(θ)*cos(φ), α*sin(θ)*sin(φ), α*cos(θ), η*cosh(ψ)),
where α = η*sinh(ψ) is the projection onto the 3D Euclidean space. These coordinates satisfy:
η² = (x² + y² + z²  w²), and 0 <= ψ < ∞, 0 <= θ <= π, 0 <= φ <= 2 π.
The coordinates of the vector γ = (α, w) satisfy: η^{2} * sinh^{2}(ψ)  η^{2} * cosh^{2}(ψ) = η^{2}.
The area of the yellow sector is η^{2} * ψ / 2. Note: ψ is not the angle ζ = arccos(w/γ) = 0.663 subtended
by the Waxis and the vector γ.
Figure 6: The radial vector α = (x, y, z) with length α is projected onto the 2D PseudoEuclidean plane
at the polar angle θ as the vector β = (x, y, 0).
The 4D space's metric is: dl² = dη² + η² dψ² + η² sinh²(ψ) [dθ² + sin²(θ) dφ²]. On a hypersurface of
radius R with dη = 0, η² = R² = x² + y² + z²  w².
The metric on the 4D "hyperbolic" hypersurface is:
dl² = R² (dψ² + sinh²(ψ) [dθ² + sin²(θ) dφ²]) where 0 <= ψ < ∞.
Figure 7: The projection of the hypersurface with η = R onto the (Z, W) plane, where θ = 0, x = y = 0.
The radius vector γ = (z, w) on the hyperbola satisfies: z²  w² = R² * sinh²(ψ)  R² * cosh²(ψ) = R².
The projection onto the Zaxis is α = R * sinh(ψ).
The distance, s, along the ψ geodesic is the line integral of the curve traced by the radius vectors from 0 to ψ (s ≠ R * ψ),
s = R * ∫_{0}^{ψ'}(cosh^{2}ψ + sinh^{2}ψ)^{1/2}dψ.
The angle ζ = arccos(w/γ).
With R = 6, and ψ = 0.6981 these values are:
z = α = R * sinh(ψ) = 4.5374; w = R * cosh(ψ) = 7.5225
γ = sqrt(z^{2} + w^{2}) = 8.7850; s = 4.8468
ζ = acos(w/γ) = 0.5428; R * ψ = 4.1888;
Yellow Area: R^{2} * ψ / 2 = 12.5664


ThreeDimensional Representations of Curvature
Hypersurface with Positive Curvature
Figure 8: (X, Y, W) space, with θ = π/2, cos(θ) = 0, sin(θ) = 1, z = 0, north pole (NP) of the
twodimensional surface (sphere) at (0, 0, 0, R).
The arc length along a ψ geodesic (φ fixed, dφ = 0) from the NP to a point = r = R * ψ.
Figure 9: (X, Y, Z) space, with ψ = π/2, cos(ψ) = 0, sin(ψ) = 1, w = 0, NP at (0, 0, R, 0).
The arc length along a θ geodesic (φ fixed, dφ = 0) from the NP to a point = r = R * θ.
Hypersurface with Negative Curvature
Figure 10: (X, Y, W) space, with θ = π/2, cos(θ) = 0, sin(θ) = 1, z = 0, pole of the twodimensional surface at (0, 0, 0, R).
The arc length along a ψ geodesic (φ fixed, dφ = 0) from the pole to the point = s = R * ∫_{0}^{ψ'}(cosh^{2}ψ + sinh^{2}ψ)^{1/2}dψ.
Figure 11: (X, Y, Z) space, with ψ = pi/4, cosh(ψ) = 1.32, sinh(ψ) = 0.87, w = 7.95, z = 5.21,
pole at (0, 0, 5.21, 7.95).
The arc length along a θ geodesic (φ fixed, dφ = 0) from the NP to a point = s = R * sinh(ψ) * θ.
Alternative Parametrizations of the Metrics
Let dΩ² = dθ² + sin²(θ) dφ², 0 <= θ <= π, 0 <= φ <= 2π.
Zero Curvature
dl² = R² (dψ² + ψ² dΩ²), let χ = ψ, then dl² = R² (dχ² + χ² dΩ²).
Let χ = ψ = r / R, → dl² = dr² + r² dΩ², 0 <= r < ∞.
Positive Curvature
dl² = R² (dψ² + sin²(ψ) dΩ²), let χ = sin(ψ) ∈ [0, 1],
dχ = cos(ψ)dψ, → dχ^{2} = cos^{2}(ψ)dψ^{2} = (1 – sin^{2}(ψ) dΨ^{2},
and dψ^{2} = dχ^{2} / (1 – sin^{2}(ψ)) = dχ^{2} / (1 – χ^{2}).
Then dl^{2} = R^{2} ( dχ^{2} / (1 – χ^{2}) + χ^{2} dΩ² ).
Let dr^{2} = R^{2} dχ^{2}/ (1  χ^{2}) → dr / R = dχ / sqrt(1  χ²) →
r / R = sin^{1}(χ) → χ = sin(r/R) →
dl^{2} = dr^{2} + R^{2} (r/R) dΩ², r = R ψ, 0 <= ψ <= π.
Negative Curvature
dl² = R² (dψ² + sinh²(ψ) dΩ²), let χ = sinh(ψ) ∈ [0, ∞),
dχ = cosh(ψ)dψ, dχ^{2} = cosh^{2}(ψ)dψ^{2} = (1 + sinh^{2}(ψ) dψ^{2},
and dψ^{2} = dχ^{2} / (1 + sinh^{2}(ψ)) = dχ^{2} / (1 + χ^{2}).
Then dl² = R² (dχ^{2} / (1 + χ^{2}) + χ^{2} dΩ²).
Let dr^{2} = R^{2} dχ^{2}/ (1 + χ^{2}) → dr / R = dχ / sqrt(1+ χ²)
→ r / R = sinh^{1}(χ) → χ = sinh(r/R)
Then dl² = dr² + R² sinh²(r/R) dΩ², r = R ψ, 0 <= ψ < ∞.
Compact Form
dl² = dr² + S_{κ}(r) dΩ² 
S_{κ}(r) = R sin(r/R) S_{κ}(r) = r S_{κ}(r) = R sinh(r/R) 
(κ = +1) (κ = 0) (κ = 1)

0 <= r < ∞ r = R ψ, 0 <= ψ <= π r = R ψ, 0 <= ψ < ∞

Standard Model
In a universe with radiation (r), matter (m), and a cosmological constant (Λ) components, the Friedmann equation can be written as:
H² / H0² = Ω^{0}_{r} / a^{4} + Ω^{0}_{m} / a^{3} + Ω^{0}_{Λ} + (1 – Ω^{0}) / a^{2},
where a = scale factor, H(t) = (da/dt) / a, H0 = H(t_{0}), t_{0} = present time, and Ω^{0}_{r}, Ω^{0}_{m}, and Ω^{0}_{Λ}, are the density parameters at t = t_{0}.
The total density of the components is Ω^{0} = Ω^{0}_{r} + Ω^{0}_{m} + Ω^{0}_{Λ}.
The curvature parameter Κ = 1, 0, 1 as Ω^{0} <, =, > 1.
The solution of the Friedmann equation provides a relation between the scale factor a and time t.
Figure 12: Model densities: Ω^{0}_{r} = 9*10^{9}, Ω^{0}_{m} = 0.31, Ω^{0}_{Λ} = Ω_{Λ} = 0.69.
Radiationmatter equality, a_{rm} = 2.9*10^{4} (Z = 3447), t_{rm} = 0.05 Myr; matterΛ equality, a_{mΛ} = 0.77
(Z = 0.2987), t_{mΛ} = 10.2 Gyr, a(t_{0}) = 1.
Model Universes – Matter and Curvature
With matter as the only component of the universe, matter’s density, Ω^{0}_{m} = Ω^{0}, determines the universe’s evolution.
The Friedmann equation can be written as: (da/dt)^{2} / H0^{2} = Ω^{0} / a + (1  Ω^{0}).
Figure 13: If Ω^{0} > 1, the universe is positively curved and collapses into a “Big Crunch.” Conversely, if Ω^{0} < 0, the universe is negatively curved and expands forever in a “Big Chill.”
Figure 14: The two trajectories only diverge significantly at a time greater than t = 1 / H0 = 14.4 Gyrs.
References
Komissarov, S.S. (2002). Cosmology. Retrieved March 1 2018, from
https://www1.maths.leeds.ac.uk/~serguei/teaching/cosmology.pdf.
Morgan, F. (1998). Riemannian Geometry: A Beginners Guide. Natick, Mass.: Peters.
Narlikar, J.V. (2002). An Introduction to Cosmology (3rd ed.). Cambridge: Cambridge UP.
Ryden, B. (2017). Introduction to Cosmology (2nd ed.). Cambridge: Cambridge UP.
Scott, Douglas (2018). Astronomy 403: Cosmology. University of British Columbia Course Notes.
Thomas, G.B. Jr. (1960). Calculus and Analytic Geometry. Don Mills, Ontario: AddisonWesley.

